Is there a name for the polynomials of the form $\sum_{i = 0}^a x^{a - i}y^i - 1$?

Question

I am trying to find out what is known about polynomials of the form $$ \sum_{i = 0}^a x^{a - i} y^i - 1 = x^a + x^{a-1}y + \dots + y^a - 1. $$ I tried searching for anything regarding the sums of the Veronese embedding into projective $n$-space, binomial expansions where all coefficients are set to one, and terms related to symmetric polynomials but came up blank. Any suggestions for things to read are greatly appreciated! This feels like the sort of polynomial that has been heavily studied.